I am trying to derive the formula for converting the differential area dxdy into dudv, where x and y are given as functions of u and v. I saw a derivation (or motivation) in a calculus book which which uses transformation and images, but I wasn't fully convinced as the motivation was not rigorous.
I tried the following, but does not seem to work out:
dx = (\partial x)/(\partial u) du + (\partial x)/(\partial v) dv.
dy = (\partial y)/(\partial u) du + (\partial y)/(\partial v) dv.
When I multiplied them, I did not get the jacobian. I am getting expressions involving du^2 and dv^2, which I do not know what to do with.
Any ideas? Please tell me what's wrong, and would be great if you send a link if you find a rigorous derivation/proof.
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